Partial Differential Equations/The Malgrange-Ehrenpreis theorem

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Vandermonde's matrix[edit]

Definition 10.1:

Let and let . Then the Vandermonde matrix associated to is defined to be the matrix

.

For pairwise different (i. e. for ) matrix is invertible, as the following theorem proves:

Theorem 10.2:

Let be the Vandermonde matrix associated to the pairwise different points . Then the matrix whose -th entry is given by

is the inverse matrix of .

Proof:

We prove that , where is the identity matrix.

Let . We first note that, by direct multiplication,

.

Therefore, if is the -th entry of the matrix , then by the definition of matrix multiplication

.

The Malgrange-Ehrenpreis theorem[edit]

Lemma 10.3:

Let be pairwise different. The solution to the equation

is given by

, .

Proof:

We multiply both sides of the equation by on the left, where is as in theorem 10.2, and since is the inverse of

,

we end up with the equation

.

Calculating the last expression directly leads to the desired formula.

Exercises[edit]

Sources[edit]